The object of this paper is to address the following question: When is a polynomial vector field on C 2 completely determined (up to affine equivalence) by the spectra of its singularities? We will see that for quadratic vector fields this is not the case: given a generic quadratic vector field there is, up to affine equivalence, exactly one other vector field which has the same spectra of singularities. Moreover, we will see that we can always assume that both vector fields have the same singular locus and at each singularity both vector fields have the same spectrum. Let us say that two vector fields are twin vector fields if they have the same singular locus and the same spectrum at each singularity.To formalize the above claim we shall prove the following: any two generic quadratic vector fields with the same spectra of singularities (yet possibly different singular locus) can be transformed by suitable affine maps to be either the same vector field or a pair of twin vector fields.We then analyze the case of quadratic Hamiltonian vector fields in more detail and find necessary and sufficient conditions for a collection of non-zero complex numbers to arise as the spectra of singularities of a quadratic Hamiltonian vector field.Lastly, we show that a generic quadratic vector field is completely determined (up to affine equivalence) by the spectra of its singularities together with the characteristic numbers of its singular points at infinity.

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of P 2 and quadratic homogeneous vector fields on C 3 , the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on C 3 having exclusively single-valued solutions.

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Consider a quadratic vector field on C 2 having an invariant line at infinity and isolated singularities only. We define the extended spectra of singularities to be the collection of the spectra of the linearization matrices of each singular point over the affine part, together with all the characteristic numbers (i.e. Camacho-Sad indices) at infinity. This collection consists of 11 complex numbers, and is invariant under affine equivalence of vector fields.In this paper we describe all polynomial relations among these numbers. There are 5 independent polynomial relations; four of them follow from the Euler-Jacobi, the Baum-Bott, and the Camacho-Sad index theorems, and are well-known. The fifth relation was, until now, completely unknown. We provide an explicit formula for the missing 5th relation, discuss it's meaning and prove that it cannot be formulated as an index theorem.

In this work we consider foliations on CP 2 which are generated by quadratic vector fields on C 2. Generically these foliations have isolated singularities and an invariant line at infinity. We say that the monodromy groups at infinity of two such foliations having the same singular points at infinity are strongly analytically equivalent provided there exists a germ of a conformal mapping at zero which conjugates the monodromy maps defined along the same loops on the infinite leaf. The object of this paper is to show that topologically equivalent generic foliations from this class must have, after an affine change of coordinates, their monodromy groups at infinity strongly analytically conjugated. As a corollary it is proved that any two such generic and sufficiently close foliations can only be topologically conjugated if they are affine equivalent. This improves, in the case of quadratic vector fields, the main result of [2] which claims that two generic, topologically equivalent and sufficiently close foliations are affine equivalent provided the conjugating homeomorphism is close enough to the identity map.